Tri-axial accelerometers and magnetometers are widely used in oil and gas well characterization to measure the components of gravitational acceleration g and the terrestrial magnetic field intensity B in a tool coordinate system.
In the context of measurement tools such as wireline-conveyed tools, g is obtained simply as (Ax, Ay, Az) where the A's are the readings of the accelerometers along the reference axes of the tool, after correction for tool acceleration. Since the tool is typically moving during a log, the accelerometer readings are low-pass filtered to remove the effects of irregular tool motion, isolating thereby the component due to gravity. The filtering can be made on the z-accelerometer readings directly. For the x- and y-accelerometers, allowance is made for tool rotation, i.e. within the filter window, the x- and y-sensors axes are not fixed relative to the gravitational field.
In the absence of stray fields caused by magnetization of tool components and material in the borehole, the magnetometer readings give B directly as (Mx, My, Mz).
By convention, the tool z-axis is the long axis of the tool, which corresponds to the borehole axis after correction for differential standoffs at the extremities of the sonde. The tool x-axis is the direction of a reference marker in the plane perpendicular to the z-axis. This reference is often called ‘P1’ or ‘Pad 1’, since in multi-pad tools, the center of the pad designated as pad 1 would be aligned to the reference marker.
The output of interest of the triaxial accelerometers and magnetometers is the tool orientation expressed in terms of the tool angles.
The orientation of the tool in the geographical coordinate system is described by the following angles (and illustrated in FIGS. 1 and 2):                Sonde Deviation (SDEV): the angle from the downward vertical to the tool z-axis (FIGS. 1 and 2);        Sonde or Hole Azimuth (HAZI): the angle in the horizontal plane from True North to the projection of the tool z-axis on the horizontal plane (FIG. 2);        Rotation of P1 from top-of-hole (a.k.a. ‘relative bearing’ RB): the angle in the xy-plane of the tool, from the top-of-hole direction to the reference P1 (FIG. 2); and        Rotation of P1 from North (P1NO): the angle in the xy-plane of the tool, from the North direction to the reference P1 (FIG. 1).        
HAZI, RB and P1NO can be undefined under the following circumstances:
1. If the hole is vertical, top-of-hole, defined as the direction of steepest ascent in the tool plane (plane perpendicular to the tool axis), is undefined. Consequently, the tool angles RB and HAZI are undefined.
2. The North direction in the tool plane is defined by the intersection of the tool plane with the vertical plane in the North-South direction. If the tool axis is horizon and pointing East or West, then the tool plane is parallel to the N-S vertical plane, so North in the tool plane, and hence P1NO, are undefined.
3. East and North are defined in terms of g and B as the directions of the vectors g×B and (g×B)×g, plus a rotation in the horizontal plane equal to the magnetic declination. At the magnetic poles, B is vertical so East and North, and consequently HAZI and P1NO, are undefined.
To avoid these singularities, it is common practice to use P1NO for subvertical, and RB for deviated and horizontal wells. With regards to large magnetic inclination, an operational limit is usually placed on the tools for magnetic latitude L<70°, i.e. I<80° (since tan I=2 tan L when the Earth magnetic field is approximated as dipolar).
The tool angles are related to the basic measurements of the tri-axial sensors according to the formulae:−SDEV=a tan 2[√(ax2+ay2),az]−HAZI=a tan 2[axmy−aymx,mz−az sin I]+D −RB=a tan 2[ay,−ax]−P1NO=a tan 2[cos D(aymz−azmy)+sin D(mx−ax sin I), cos D(azmx−axmz)+sin D(my−ay sin I)]where(ax,ay,az)=(Ax,Ay,Az)/√(Ax2+Ay2+Az2),(mx,my,mz)=(Mx,MyMz)/√(Mx2+My2+Mz2),
D is the magnetic deviation that must be known a priori, and sin I is the sine of the magnetic inclination measured as: sin I=axmx+aymy+azmz.
The ideal tri-axial instrument would have sensors that
1. output a reading equal to the field component parallel to the sensor axis, and
2. are aligned perfectly in parallel with the orthogonal tool reference axes.
In practice, a given instrument must be calibrated to eliminate systematic deviations from the ideal. If the response of a sensor is assumed to be linear, then each sensor in a triaxial system is characterized by four calibration parameters:                a scaling factor,        an offset (bias),        two misalignment angles giving the deviation of the sensor axis from the two tool axes to which it is supposedly at 90°.        
At a given temperature, the parameters can be determined using the Total Field Method of Estes and Walters (Estes, R. and Walters, P., 1969, ‘Improvement of Azimuth Accuracy by Use of Iterative Total Field Calibration Technique and Compensation for System Environment Effects’, SPE 19546). For a tool consisting of triaxial accelerometers and magnetometers, there are 24 calibration parameters at a given temperature. A full calibration determines these parameters at several temperatures, covering the operational temperature range of the tool, so that their variation with temperature can be accounted for by interpolation. A full calibration is carried out at surface periodically. More frequently, a calibration ‘check’ is made at ambient temperature to ensure that the tools remain in calibration.
The Total Field Method is applied independently to the triaxial accelerometers and magnetometers. In an environment where the field F (g or B) is uniform and constant, and the magnitude |F| is known, the triaxial sensors (accelerometers or magnetometers) are oriented in a number of positions so that each individual sensor is exposed to field values covering the entire range of possible values from −|F| to +|F|. These positions are conveniently obtained by rotating the tool about its x-, y- and z-axes, with the rotation axis aligned approximately East-West. At each position, the corrected measured field components and |F| can be computed using trial values of the calibration parameters. The true values of the calibration parameters are then determined as the values which:
1. minimize the rms difference between the |F|Corrected values and the known value of |F|, and
2. render FαCorrected=constant, for the points obtained by rotation about the α-axis.
After calibration, the remaining uncertainty in the measurements can be characterized by the standard deviation, respectively σA and σM for the individual accelerometers and magnetometers. Typical values for σA and σM are 0.02 m/s2, and 150 nT respectively at a frequency of 15 Hz and inclusive of electronics noise, for avionic grade instruments. For g=9.80 m/s2 and B˜50000 nT, the uncertainties in the measured direction of g and B would be 0.1° and 0.2° respectively as indicated by the schematic in FIG. 3.
In FIG. 3, the thick arrow represents the field vector g or B. In three dimensions, the measured values of the vector falling within σ of the true value are enclosed in a sphere (shown in dotted line) with radius σ. The standard deviation in the direction of the vector would therefore be equal to the half-angle of the cone C subtended by the sphere. In the case of g, the half angle of the cone is σA/|g|=0.002 radians, or 0.1°.
In fact when logging, the magnetometer measurements are typically averaged over four samples, and the accelerometer measurements are effectively averaged over 36 samples in the lowpass filter to remove tool acceleration. Taking the averaging into account, we can write, for typical wireline logs:σA=(σA/g)/√36˜0.0003σ*M=(σM/B)/√4˜0.0015where the σ*'s are expressed as fractions of the field magnitudes.
Various methods have been proposed for correcting accelerometer and magnetometer measurements. For example, U.S. Pat. No. 6,179,067 describes methods for determining magnetometer errors during wellbore survey operations in which a model is used to correct the observed data and the corrected data are transformed from the tool coordinate system to a different coordinate system referenced to the earth. The difference between the corrected transformed data and reference data in the earth coordinate system is minimized to determine the model parameters.
The present invention provides a procedure that can be used to quality control and correct for offsets in the wireline triaxial accelerometer and magnetometer measurements in situ. i.e. using data acquired during a log to correct the offsets as they existed during the log.